SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1248-A1272, January 2021.
In this paper, two finite difference numerical schemes are proposed and analyzed for the droplet liquid film model with a singular Lennard--Jones energy potential involved. Both first and second order accurate temporal algorithms are considered. In the first order scheme, the convex potential and the surface diffusion terms are updated implicitly, while the concave potential term is updated explicitly. Furthermore, we provide a theoretical justification that this numerical algorithm has a unique solution such that the positivity is always preserved for the phase variable at a pointwise level, so that a singularity is avoided in the scheme. In fact, the singular nature of the Lennard--Jones potential term around the value of 0 prevents the numerical solution reaching such a singular value, so that the positivity structure is always preserved. Moreover, an unconditional energy stability of the numerical scheme is derived, without any restriction for the time step size. In the second order numerical scheme, the BDF temporal stencil is applied, and an alternate convex-concave decomposition is derived, so that the concave part corresponds to a quadratic energy. In turn, the combined Lennard--Jones potential term is treated implicitly, and the concave part is approximated by a second order Adams--Bashforth explicit extrapolation, and an artificial Douglas--Dupont regularization term is added to ensure the energy stability. The unique solvability and the positivity-preserving property for the second order scheme could be similarly established. In addition, optimal rate convergence analysis is provided for both the first and second order accurate schemes. A few numerical simulation results are also presented, which demonstrate the robustness of the numerical schemes.

中文翻译：

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液体薄膜粗化模型的结构保持，能量稳定数值方案

SIAM科学计算杂志，第43卷，第2期，第A1248-A1272页，2021年1月。
本文针对奇异的Lennard-Jones势能提出了两种有限差分数值方案，并对液滴液膜模型进行了分析。一阶和二阶精确时间算法均被考虑。在一阶方案中，隐式更新凸电位和表面扩散项，而显式更新凹电位。此外，我们提供了一种理论上的证明，即该数值算法具有独特的解决方案，可以始终将相位变量的正性保持在逐点水平，从而避免了该方案中的奇异性。实际上，Lennard-Jones势项在0值附近的奇异性质阻止了数值解达到这样的奇异值，从而始终保持阳性结构。此外，导出了数值方案的无条件能量稳定性，而对时间步长没有任何限制。在二阶数值方案中，应用了BDF时间模版，并进行了交替的凸凹分解，因此凹部对应于二次能量。反过来，隐式处理组合的Lennard-Jones势项，并通过二阶Adams-Bashforth显式外推法近似凹入部分，并添加了人工Douglas-Dupont正则化项以确保能量稳定性。可以类似地建立二阶方案的唯一可溶性和正性保留性质。另外，为一阶和二阶精确方案都提供了最佳速率收敛分析。